<h2>The ring of Siegel modular forms of degree 2 with respect to the 
<a class="knowl-title" knowl="mf.siegel.group.symplectic">full modular group</a></h2>

<div class="literature">
  <ul>
    <li><span class="name">J.-I. Igusa:</span> On Siegel modular forms of genus two. Amer. J. Math. 84 (1962), 175-200, <a href="http://www.ams.org/mathscinet-getitem?mr==0141643">MR0141643</a></li>
  </ul>
</div>

<p>
By a classical result of <span class="name">Jun-Ichi Igusa</span> (On Siegel modular forms of genus two. Amer. J. Math. 84 (1962), 175-200, <a href="http://www.ams.org/mathscinet-getitem?mr==0141643">MR0141643</a>), the ring <script type="math/tex">M_{2*}({\rm Sp}(4,\mathbb{Z}))</script> of Siegel modular forms of degree 2 with <b>even weights</b> with respect to 
the 
<a class="knowl-title" knowl="mf.siegel.group.symplectic">full modular group</a>
$Sp(4,{\Bbb Z})$ is generated by the following four algebraically independent
<a class="knowl-title" knowl="mf.siegel.eisenstein_series">Eisenstein series</a>:
</p>

<ul>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Sp4Z', form = 'E', weight = 4, page = 'specimen') }}">$\psi_4$</a>,
</li><li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Sp4Z', form = 'E', weight = 6, page = 'specimen') }}">$\psi_6$</a>,
</li><li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Sp4Z', form = 'E', weight = 10, page = 'specimen') }}">$\psi_{10}$</a>,
</li><li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Sp4Z', form = 'E', weight = 12, page = 'specimen') }}">$\psi_{12}$</a>,
</li></ul>

Alternatively, \(\psi_{10}\) can be replaced by the cusp form

<ul><li> 
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Sp4Z', form = 'Maass', weight = 10, page = 'specimen') }}">$\chi_{10}$</a>
$=-43867\cdot 2^{-12}3^{-5}5^{-2}7^{-1}53^{-1}(\psi_4\psi_6-\psi_{10})$
</li></ul>

and \(\psi_{12}\) can be replaced by the cusp form

<ul><li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Sp4Z', form = 'Maass', weight = 12, page = 'specimen') }}">$\chi_{12}$</a>
$=131\cdot593\cdot2^{-13}3^{-7}5^{-3}7^{-2}337^{-1}(441\psi_4^3+250\psi_6^2-691\psi_{12}).$
</li></ul>

<p>
Note that \(\chi_{10}\) is a Saito-Kurokawa lifting coming from the cuspform of weight \(10\) for \({\rm SL}(2,\mathbb{Z})\), and \(\chi_{12}\) is a Saito-Kurokawa lifting coming from the cuspform of weight \(12\).
</p>

<p>
The even weight forms are:
$${\Bbb C}[\psi_4,\psi_6,\chi_{10},\chi_{12}],$$
with the cusp forms being the ideal generated by $\chi_{10}, \chi_{12}$.

To generate the odd weight forms, there is one more generator, namely
<a href="{{ url_for('not_yet_implemented') }}">$\chi_{35}$</a>.
For a formula for $\chi_{35}$, see [references...]
</p>

<p>
The odd weight forms are:
$$\chi_{35}\,{\Bbb C}[\psi_4,\psi_6,\chi_{10},\chi_{12}],$$
where $\chi_{35}$ is a weight 35 cusp form.
</p>
